![\"\"](\"https:\/\/wollen.org\/blog\/wp-content\/uploads\/2026\/08\/linear_example.png\")
<\/a>You don’t need any technical definitions to know the linear model is misspecified. The relationship between X and Y obviously follows a curved path.<\/p>\n
Rather than a simple linear regression of this form:<\/p>\n
We can use SciPy to calculate coefficients A<\/strong>, B<\/strong>, and C<\/strong> and fit a model. Of course a real statistician would use Greek symbols and better notation but it really isn’t necessary. It boils down to a parabola equation. Occasionally you’ll want a higher-order polynomial but the approach is the same.<\/p>\n Our college football spread-win% data is nonlinear, similar to the randomly generated data above. Let’s work on building a model.<\/p>\n Start by loading the dataset into a pandas DataFrame.<\/p>\n The output:<\/p>\n So each row represents one football game. We’ll have to somehow group by spread<\/em> and then check how often the favorite wins at each value.<\/p>\n For this analysis let’s filter our DataFrame down to the most recent 20 seasons. The game has changed a lot in 20 years but it’s changed even more since 1978. 20 years will leave us plenty of data to work with.<\/p>\n Let’s ignore zero-point “pick’em” spreads because those are obviously 50\/50 games. Let’s also cap the regression at 14-point favorites. Beyond that mark there isn’t enough data.<\/p>\n We’re left with 10,375 games. That will be plenty.<\/p>\n Now we need a column to tell us whether the favorite won the game. Apply<\/em> a custom function row-wise.<\/p>\n The output is below. You can see the favorite won in four of the five most recent games.<\/p>\n If your instinct is to The output:<\/p>\n Now we can create a win percentage column using the True<\/em> and False<\/em> columns. This will tell us how often a 1-point favorite wins, and a 1.5-point favorite, and so on up to 14.<\/p>\n One more time, the output is below. This passes the smell test in my opinion. Small favorites win a little more than 50% of the time but there’s a lot of noise in the data.<\/p>\n We need a nonlinear regression line for our scatter plot. SciPy makes this easy with its The method returns values for each coefficient as well as a covariance matrix NumPy array. We won’t be using the array so assign it to a throwaway underscore.<\/p>\n That’s all there is to it. We can use these three values to plot the curve defined above:<\/p>\n Before moving on to plotting, let’s create x and y regression line lists.<\/p>\n The Matplotlib code is straightforward and I’ve covered it 50 times before on the blog, so I’ll keep this brief. I use a custom Matplotlib style that will be linked at the bottom of this post.<\/p>\n We have a scatter plot using It’s very blue! To answer the original question, a 7-point favorite wins about 69% of the time in college football<\/strong>.<\/p>\n The important point is that there are diminishing returns to win% as point spread grows larger. The difference between a 3- and 4-point favorite is bigger than the difference between 13 and 14. That’s why we couldn’t do a simple linear regression and draw a straight line through the data.<\/p>\n Remember how we excluded pick’em spreads from the regression. According to our model there is no measurable difference between a 1-point favorite and a 1-point underdog. They’re all coinflip games. It makes sense but it’s not something I would have expected. To be honest I’d still prefer my team to be favored.<\/p>\n I won’t post the code but I want to show the same analysis for college basketball. I included moneyline conversions in the upper-left corner.<\/p>\n The correlation is stronger despite using roughly the same amount of data. It’s fair to say there are fewer surprises in basketball.<\/p>\n<\/a>where we calculate slope (m<\/strong>) and intercept (b<\/strong>), we need a nonlinear model of this form:<\/p>\n<\/a><\/p>\n
\n1. Prepare the data.<\/h4>\n
import pandas as pd\r\n\r\npd.set_option(\"display.expand_frame_repr\", False) # Improves printing to screen.\r\n\r\ndf = pd.read_csv(\"cfb_spreads_1978-2025.csv\", parse_dates=['date'])\r\n\r\nprint(df.head())<\/pre>\n
date season away home score_away score_home whos_favored spread\r\n0 1978-09-01 1978 Penn State Temple 10 7 away 24.5\r\n1 1978-09-02 1978 Arkansas State Tulsa 20 21 home 1.0\r\n2 1978-09-02 1978 Nebraska Alabama 3 20 home 11.5\r\n3 1978-09-02 1978 West Texas A&M Mississippi State 0 28 home 12.0\r\n4 1978-09-09 1978 Florida State Syracuse 28 0 away 1.5<\/pre>\n
df = df[df['season'] >= 2006]\r\n\r\ndf = df[(df['spread'] >= 1) & (df['spread'] <= 14)]<\/pre>\n
def check_if_fav_wins(row):\r\n if row['whos_favored'] == \"away\":\r\n return row['score_away'] > row['score_home']\r\n else:\r\n return row['score_home'] > row['score_away']\r\n\r\n\r\ndf['fav_wins'] = df.apply(check_if_fav_wins, axis=1)\r\n\r\nprint(df.tail())<\/pre>\n
date season away home score_away score_home whos_favored spread fav_wins\r\n29612 2026-01-02 2025 Wake Forest Mississippi State 43 29 home 3.0 False\r\n29613 2026-01-02 2025 Arizona SMU 19 24 home 2.5 True\r\n29615 2026-01-08 2025 Miami (FL) Ole Miss 31 27 away 3.0 True\r\n29616 2026-01-09 2025 Oregon Indiana 22 56 home 3.0 True\r\n29617 2026-01-19 2025 Miami (FL) Indiana 21 27 home 7.5 True<\/pre>\n
groupby<\/code> the spread<\/em> column and go from there, it’s a good thought and you could<\/span> make that work. But there’s a better option. When we want to combine values of one column and count multiple values within another column, crosstab<\/code> is the perfect tool.<\/p>\ndf2 = pd.crosstab(df['spread'], df['fav_wins'])\r\n\r\nprint(df2.head())<\/pre>\n
fav_wins False True \r\nspread \r\n1.0 176 177\r\n1.5 165 224\r\n2.0 138 128\r\n2.5 344 379\r\n3.0 405 546\r\n<\/pre>\n
df2['win_pct'] = df2[True] \/ (df2[True] + df2[False]) * 100\r\n\r\nprint(df2.head())<\/pre>\n
fav_wins False True win_pct\r\nspread \r\n1.0 176 177 50.141643\r\n1.5 165 224 57.583548\r\n2.0 138 128 48.120301\r\n2.5 344 379 52.420470\r\n3.0 405 546 57.413249<\/pre>\n
\n2. Fit a model.<\/h4>\n
curve_fit<\/code> method. We just have to define a function and pass it into curve_fit<\/code> along with the data.<\/p>\nfrom scipy.optimize import curve_fit\r\n\r\n\r\ndef model(x, a, b, c):\r\n return a * x**2 + b * x + c\r\n\r\n\r\n(coef_a, coef_b, coef_c), _ = curve_fit(f=model,\r\n xdata=df2.index,\r\n ydata=df2['win_pct'])<\/pre>\n
<\/a><\/p>\nx_reg = df2.index.tolist()\r\ny_reg = [coef_a * x**2 + coef_b * x + coef_c for x in x_reg]<\/pre>\n
\n3. Plot the data.<\/h4>\n
df2<\/code> data and the regression line is drawn on top. I also include a horizontal dotted line representing a 50% winning percentage. The regression line equation is displayed at the top of the figure using text()<\/code>.<\/p>\nimport matplotlib.pyplot as plt\r\n\r\nplt.style.use(\"wollen_cfb.mplstyle\")\r\nfig, ax = plt.subplots()\r\n\r\nwhite = \"#F1F1F1\"\r\nyellow = \"#FFD139\"\r\ngray = \"#444\"\r\n\r\nax.scatter(df2.index,\r\n df2['win_pct'],\r\n color=white,\r\n ec=gray,\r\n lw=0.6,\r\n zorder=2)\r\n\r\nax.plot(x_reg,\r\n y_reg,\r\n color=yellow,\r\n zorder=3)\r\n\r\nax.plot([-100, 100],\r\n [50, 50],\r\n color=white,\r\n linestyle=\"--\",\r\n lw=0.8,\r\n zorder=1)\r\n\r\nx_ticks = range(15)\r\ny_ticks = range(45, 95, 5)\r\n\r\nax.set(xticks=x_ticks,\r\n xticklabels=[f\"-{n}\" if n != 0 else \"PK\" for n in x_ticks],\r\n xlim=(-0.3, 14.3),\r\n yticks=y_ticks,\r\n yticklabels=[f\"{n}%\" for n in y_ticks],\r\n ylim=(44.5, 90.5))\r\n\r\nfirst_month = df['date'].min().strftime('%b %Y')\r\nfinal_month = df['date'].max().strftime('%b %Y')\r\nax.set_title(f\"NCAA Football \u2022 Spread and Win % \u2022 {first_month} \u2013 {final_month} \u2022 n={df.shape[0]:,}\")\r\n\r\nax.text(x=7,\r\n y=89.5,\r\n s=f\"y = {coef_a:.2f}x\u00b2 + {coef_b:.2f}x + {coef_c:.2f}\",\r\n size=11,\r\n weight=\"500\",\r\n ha=\"center\",\r\n va=\"top\",\r\n bbox={\"fc\": \"#005EB8\", \"ec\": \"None\"})\r\n\r\nplt.savefig(\"cfb_spread_regression.png\", dpi=200)<\/pre>\n
\n4. The output.<\/h4>\n
<\/a><\/p>\n
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